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To be sure that \( g \) really is right adjoint to \( f \) in Example 1.97 [below], there are twelve things to check; do so. That is, for every \( p \in P \) and \( q \in Q \), check that \( f(p) \le q \text{ iff } p \le g(q) \).

Let \( g \) be the map that preserves labels, and let \( f \) be the map that preserves labels as far as possible but sends \( f(3.9) = 4 \).

## Comments

Using Proposition 1.93, we can check just 7 things:

$$ \begin{array}{c|c|c|c} p&f(p)&g(f(p))&p \leq g(f(p))\\ \hline 1 &1&1&T\\ 2 &2&2&T\\ 3.9&4&4&T\\ 4 &4&4&T\\ \end{array} $$ and

$$ \begin{array}{c|c|c|c} q&g(q)&f(g(q))&f(g(q)) \leq q\\ \hline 1&1&1&T\\ 2&2&2&T\\ 4&4&4&T\\ \end{array} $$

`Using Proposition 1.93, we can check just 7 things: \[ \begin{array}{c|c|c|c} p&f(p)&g(f(p))&p \leq g(f(p))\\\\ \hline 1 &1&1&T\\\\ 2 &2&2&T\\\\ 3.9&4&4&T\\\\ 4 &4&4&T\\\\ \end{array} \] and \[ \begin{array}{c|c|c|c} q&g(q)&f(g(q))&f(g(q)) \leq q\\\\ \hline 1&1&1&T\\\\ 2&2&2&T\\\\ 4&4&4&T\\\\ \end{array} \]`